Dimension reduction is a key algorithmic tool with many applications including nearest-neighbor search, compressed sensing and linear algebra in the streaming model. In this work we obtain a sparse version of the fundamental tool in dimension reduction --- the Johnson--Lindenstrauss transform. Using hashing and local densification, we construct a sparse projection matrix with just O~(ε1) non-zero entries per column. We also show a matching lower bound on the sparsity for a large class of projection matrices. Our bounds are somewhat surprising, given the known lower bounds of Ω(ε21)
Nearby in the stack
both on the number of rows of any projection matrix and on the sparsity of projection matrices generated by natural constructions. Using this, we achieve an
O~(ε1)
update time per non-zero element for a
(1±ε)
-approximate projection, thereby substantially outperforming the
O~(ε21)
update time required by prior approaches. A variant of our method offers the same guarantees for sparse vectors, yet its
O~(d)
worst case running time matches the best approach of Ailon and Liberty.